Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R²

2007/04/20 by J. Colliander, M. Keel, Colliander, J. +10
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Numerical methods for differential equations #math-ph #math.AP #math.MP #msc:35Q55

paper · pdf · doi:10.48550/arxiv.0704.2730

arxiv created 2007/04/20 · arxiv updated 2009/12/01

Abstract

The initial value problem for the cubic defocusing nonlinear Schrödinger equation i ∂_t u + Δu = |u|² u on the plane is shown to be globally well-posed for initial data in H^s (\R²) provided s>1/2. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise.

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